The William Lowell Putnam Mathematical Competition

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The William Lowell Putnam Mathematical Competition

• Dana Lecture Series
• Jan 18th 2006
• Dan McQuillan

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The William Lowell Putnam Mathematical
CompetitionOutline

• History of the contest
• The rules and rewards
• Examples of questions

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The William Lowell Putnam Mathematical Competition

• 1882 Mr. Putnam attends Harvard
• 1921 He writes a letter to Harvard Graduates
  Magazine
• 1927 Elizabeth Lowell Putnam establishes a
  125,000 trust in her will
• 1928 First attempt at a contest takes place
  between Harvard and Yale-in English

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The William Lowell Putnam Mathematical Competition

• 1933 Contest between Harvard and West Point in
  mathematics
• 1933 Harvard president, Abbott Lawrence Lowell,
  retires
• 1935 Elizabeth Lowell Putnam dies, leaving
  control of the contest to her sons George and
  August

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The William Lowell Putnam Mathematical Competition

• 1938 Consultant G. D. Birkhoff of Harvard decides
  to open the contest to any 3 person team in the
  United States or Canada
• 1938 University of Toronto wins the first contest
• 1939 University of Toronto is disqualified for
  having won the previous year

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The William Lowell Putnam Mathematical Competition

• 1943-45 World War II prevents the contest from
  operating
• 1946 Creation of a committee of prominent
  mathematicians, G. Polya, T. Rado and I.
  Kaplansky to write the questions, putting
  emphasis on questions which require ingenuity
• 1946 Previous years winner may compete

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Putnam Fellows

• Irving Kaplansky 1938
• Richard Feynman 1939
• John Milnor 1949 and 1950
• Kenneth G. Wilson 1954 and 1956
• David Mumford 1955 and 1956
• Daniel Quillen 1959

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Putnam Fellows

• Donald Passman 1959
• Neil Koblitz 1968
• Karl Rubin 1974
• Adam Logan 1992 and 1993
• Ravi Vakil 1988, 1989, 1990 and 1991

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Individual Rules

• No person with a college degree can enter the
  contest
• No person may enter the contest more than 4 times
• 3 hours to answer questions A1 through A6
• 2 hours for lunch
• 3 hours to answer questions B1 through B6
• No calculators

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Contest Rules

• Each question is worth 10 points
• Partial credit will only be given for significant
  and substantial progress towards a solution
• 1 complete and correct solution is worth more
  than 3 partially correct solutions

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Other facts

• The median score is almost always less than 10
  and it is often 0
• A list of the top 500 contestants is sent to all
  participating institutions
• A list of the top 100 contestants is published in
  the journal, American Mathematical Monthly
  usually in October

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Who cares?

• Graduate schools in mathematics!
• The professional mathematics community
• Individuals who have competed at any time in
  their life
• Any school that wants its best students exposed
  to the same questions that world class
  institutions are showing their students

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William Lowell Putnam

• It seems probable that the competition which has
  inspired young men to undertake and undergo so
  much for the sake of athletic victories might
  accomplish some result in academic fields.

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Kedlaya, Poonen and Vakil

• Be patient when working on a problem. Learning
  comes more from struggling with problems than
  from solving them...Most students, when they
  first encounter Putnam problems, do not solve
  more than a few, if any at all, because they give
  up too quicklyproblem-solving becomes easier
  with experience it is not a function of
  cleverness alone.

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Examples of Recent Questions
2002 A2 Given any five points on a sphere, show
that some four of them must lie on a closed
hemisphere.
Solution Consider a great circle through any
pair of points. The sphere is split by it into
two closed hemispheres overlapping on this
circle.
Of the three remaining points, at least two must
live in one of these hemispheres. That hemisphere
has at least 4 points.
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2003 A1
Let n be a fixed positive integer. How many ways
are there to write n as a sum of positive
integers,
with k an arbitrary positive integer and
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What?

• Try n4
• 431 does not work
• 413 does not work
• These work
• 4, 22, 112, 1111

k arbitrary
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2003 A1. Example n5

• 55
• 54 no
• 53no
• 523
• 5122
• 51112
• 511111

k arbitrary
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2003 A1 representing integers

• If you write the answer is n, you will get 1
  point out of 10 for this question.
• Key insight change notation to reflect the most
  important feature of the examples
• Let and let r be the number of
  
• s such that .

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2003 A1 Solution

• There are k summands, so
• i.e
  and
• The number of such expressions is n, one for each
  possible value of k between 1 and n

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2003 A1. Example n5

• 55
• 54 no
• 53no
• 523
• 5122
• 51112
• 511111

(1)(5)0
(2)(2)1
(3)(1)2
(4)(1)1
(5)(1)0
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A preparation question

• Given any 51 numbers from the set 1,2,3,,100,
  prove that one must divide another.
• We proved this in two ways. First, we found a
  conjecture for a more general result.
• Given n1 numbers between 1 and 2n, prove that
  one must divide another. (Solution 1 was quite
  involved).

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A preparation question

• Solution 2 Here are the first 100 numbers,
  reorganized into Boxes to fit our problem
• 1,2,4,8,16,32,64---1st box
• 3,6,12,24,48,96---2nd box
• 5,10,20,40,80---3rd box
• 7,14,28,56---4th box
• Etc
• There are 50 boxes. If you pick 2 numbers from
  the same box, then youre done. If you pick 51 or
  more, then youve got to pick two from the same
  box.

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2005 A1 Problem

• Show that every positive integer is a sum of one
  or more numbers of the form , where
  are nonnegative integers and no summand
  divides another

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2005 A1 Try a few examples

• 11
• 22
• 33, 44, 66, 88, 99, 1212, 1616
• 532
• 743
• 1064 (compare expression 532)
• For a tough example, dont pick an even number
• 15123 does not work
• 1596

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2005 A1 How to express 61?
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9 3
1 54
18 6
2
36 12
4

24 8

48 16


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2005 A1 solution

• By way of contradiction, assume there is a
  counterexample, and let be the smallest
• Note that cannot have a factor of 2 (why?)
• Similarly cannot have a factor of 3
• Now assume some
• Since is not even, we may write
• for some
• .

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2005 A1 solution

• By the minimality of , has a suitable
  representation, say
• But,
• Now we have a contradiction since we have found a
  suitable representation for (some
  explanation is still required).

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Proof of lemma
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2004 B2

• Let m and n be positive integers. Show that

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2005 B2

• Find all positive integers
• such that
• and
• Lemma Let be positive real
  numbers
• such that Then,
• By the lemma,

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Kedlaya, Poonen and Vakil

• Be patient with the solutions as well.
  Mathematics is meant to be read slowly and
  carefully. If there are some steps in a solution
  that you do not follow, try discussing it with a
  knowledgeable friend or instructor. Most research
  mathematicians do the same when they are stuck
  (which is most of the time)We hope that you
  follow up on the ideas that interest you most.

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References

• 1. The William Lowell Putnam Mathematical
  Competition problems and solutions 1938-1964, by
  A.M. Gleason, R.E. Greenwood and L.M. Kelly
• 2. The William Lowell Putnam Mathematical
  Competition 1985-2000, by K. Kedlaya, B. Poonen
  and R. Vakil
• 3. Problem-solving through problems, by L. Larson
• 4. http//math.scu.edu/putnam/index.html
• 5 http//www.d.umn.edu/jgallian/putnamfel/PF.html

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                                              Wi
lliam Lowell Putnam
Thank you Mr. Putnam!